Canon

Elena Gilbert met Stefan Salvatore's charming and vindictive older brother Damon Salvatore in the second episode of the television series. They have saved each other's lives so many times, it practically deserves its own summary.

From the start of the series, Damon stirred up trouble between Elena and Stefan. Elena had every reason to hate Damon after he killed Vicki Donovan and others, but somehow she ended up caring for him. She believed he could be a better man. Through season one, the two forged an unlikely friendship. At the end of the season one finale 'Founder's Day', Damon encountered Elena on the porch of her house. He thanked her for deeming him worth saving and kissed her, not knowing that he was kissing Katherine (Elena's vampire doppelganger, Damon's sire).

Elena's feelings for Damon grew more conflicted in season two. When Klaus Mikaelson and his brother Elijah came to Mystic Falls intent on using Elena in their magic ritual, Damon was esperate to save Elena's life. While doing so, he got bit by a werewolf and while he was on his death bed, Elena kissed him.

End of season three was when Elena was forced to make a choice, which brother she wanted to see before they died and she chose Stefan. While she was on her way to see him, Matt Donovan (who was driving the car she was in), drove the car off a bridge to avoid colliding with Rebekah. As a result, Elena died and due to vampire blood in her system, she turned into a vampire.

For the majority of season 4, Elena was sired to Damon, compelled to follow his orders. When she turned off her emotions, Damon was unable to use their connection to get her to feel again, so he killed Matt in front of her, making her feel the grief was her friend. At the end of season four, unsired Elena chose Damon as the man she loved.

Fandom & Fanon

more common tropes in fanworks, additional fandom info, especially in later seasons

People began shipping Damon/Elena since the book series that inspired the TV Show. Delena is one of the shows most popular relationships.

It's worth pointing out that not all fans ship Damon and Elena. Before the pairing became canon, some fans (Stelena shippers among them) argued that since Elena said "no" to Damon on a number of occassions, she shouldn't be pressured into "giving him a chance."

There seems to be this idea that her no will become a maybe, that because Damon loves her, because he “tries”, she owes him the benefit of “giving him a chance.” This is bullshit. Complete and total bullshit. She owes him nothing. She doesn’t owe him an explanation. She doesn’t owe him time. She doesn’t owe him a chance. She owes him jack shit. Women do not owe every man who wants to toss his hat in the ring a chance. They just don’t. And saying “no, I’m not interested” doesn’t make you mean, doesn’t make you bad, doesn’t make you a bitch. You are not cold-hearted, your vagina is not frozen, you are not frigid or a prude. You are a person, and you have the right to say who you want to have a relationship with. Women are not some prize to be traded around.[1]

When Elena became a vampire it was also revealed that she was sired to Damon. A lot of fans (Delena shippers being the majority) ignored the consent issues connected with that fact, instead, choosing to enjoy the ship and the characters as they were presented. The sire bond plays little to no role in fanworks.

However, some fans of the show saw the relationship that developed under the sire bond as non-con or dub-con at best.

whatever else their flaws, the writers have been super consistent about Damon's character and his abuse, misogyny and heavy-handed dehumanization of [Elena].
So they solved this problem by making her physically incapable of disagreeing with him, an emotional puppet with her thoughts and her emotions under control. Jesus fucking Christ, she vomits blood when he disagrees with what she's eating.[2]